
Here, \(m_{12}\)  depends on the mixing angle \(\beta \)  between \(v1\)  and \(v2\)  and is defined as \(tan~ \beta =v2/v1\) .
While \(tan~\beta \)  is a physical parameter here, it is still possible to redefine the two
doublets and go to a basis of the two doublets:
the so-called Higgs basis (H1 and H2),see  [1]}. The quartic coupling constants are real for our scenario and hence \(\lambda _5^*=\lambda _5\) . The explicit expressions for the couplings are given by:
\(\lambda _1 &= \frac{1}{v_{\rm sm}^2 \cos ^2 \beta } \left( - \mu ^2 \tan \beta + m_h^2 \sin ^2 \alpha + m_H^2 \cos ^2\alpha \right) \\\lambda _2 &= \frac{1}{v_{\rm sm}^2 \sin ^2 \beta } \left( - \frac{\mu ^2}{\tan \beta } + m_h^2 \cos ^2 \alpha + m_H^2 \sin ^2\alpha \right) \\\lambda _3 &= \frac{1}{v_{\rm sm}^2 } \left( - \frac{2 \mu ^2}{\sin 2\beta } + 2 m_{H_\pm }^2 + \left(m_H^2 - m_h^2 \right)\frac{\sin 2 \alpha }{\sin 2 \beta }\right) \\\lambda _4 &= \frac{1}{v_{\rm sm}^2 } \left( \frac{2 \mu ^2}{\sin 2\beta } + m_A^2 - 2 m_{H_\pm }^2 \right) \\\lambda _5 &= \frac{1}{v_{\rm sm}^2 } \left( \frac{2 \mu ^2}{\sin 2\beta } - m_A^2 \right)\) 
